- #1

mananvpanchal

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Hello,

Suppose, There is two points [itex][t_{a1}, x_{a1}][/itex] and [itex][t_{a2}, x_{a2}][/itex] stationary in my frame A. I say [itex]L_a=x_{a2}-x_{a1}[/itex].

If I want to find x component of co-ordinates in other frame B which is moving relative to me with constant speed v.

I have to use the equation

1. [itex]x_b=\gamma(x_a-vt_a)[/itex], where [itex]\gamma > 1[/itex].

So, I can get [itex]x_{b1}=\gamma(x_{a1}-vt_{a1})[/itex], [itex]x_{b2}=\gamma(x_{a2}-vt_{a2})[/itex].

If I take [itex]t_{a1}=t_{a2}=0[/itex], then [itex]x_{b1}=\gamma x_{a1}[/itex] and [itex]x_{b2}=\gamma x_{a2}[/itex].

Now, how much distance B measures in his co-ordinate system between this two points is [itex]x_{b2} - x_{b1} = \gamma (x_{a2} - x_{a1})[/itex].

If B says [itex]L_b=x_{b2} - x_{b1}[/itex].

So, I would get [itex]L_b=\gamma L_a[/itex].

Here, [itex]L_b > L_a[/itex]. But, this should not be the case.

If I want length contraction then I have to derive it oppositely. (I found this method from http://en.wikipedia.org/wiki/Length_contraction#Derivation and http://www.fourmilab.ch/etexts/einstein/specrel/www/#SECTION14)

I have to pick some other B's co-ordinates [itex][t_{b1}, x_{b1}][/itex] and [itex][t_{b2}, x_{b2}][/itex] which is stationary in B's frame. B says [itex]L_b=x_{b2}-x_{b1}[/itex]

I have to use this equation

2. [itex]x_{a}=\gamma(x_{b}-vt_{b})[/itex], where [itex]\gamma > 1[/itex].

I would find [itex]x_{a1}=\gamma(x_{b1}-vt_{b1})[/itex] and [itex]x_{a2}=\gamma(x_{b2}-vt_{b2})[/itex].

If I take [itex]t_{b1}=t_{b2}=0[/itex] then, [itex]x_{a1}=\gamma x_{b1}[/itex] and [itex]x_{a2}=\gamma x_{b2}[/itex].

So, [itex]x_{a2} - x_{a1} = \gamma (x_{b2} - x_{b1})[/itex].

I says [itex]L_a=x_{a2}-x_{a1}[/itex].

so, [itex]L_a=\gamma L_b[/itex].

so, [tex]L_b=\frac{L_a}{\gamma}[/tex]

so, [itex]L_b < L_a[/itex]. Now, we can say this as length contraction.

But, I have started this derivation using B's co-ordinates. I as A don't know B's co-ordinates. I only know my co-ordinates because I can physically define it. I have to calculate B's co-ordinates to get Length Contraction.

I cannot use equation (1) for that, it wouldn't give me length contraction.

How can I get B's co-ordinates using my own?

If I as A has a stationary point [itex][t_a, x_a][/itex], I can calculate B's co-ordinate using my own in Galilean transformation.

Suppose, There is two points [itex][t_{a1}, x_{a1}][/itex] and [itex][t_{a2}, x_{a2}][/itex] stationary in my frame A. I say [itex]L_a=x_{a2}-x_{a1}[/itex].

If I want to find x component of co-ordinates in other frame B which is moving relative to me with constant speed v.

I have to use the equation

1. [itex]x_b=\gamma(x_a-vt_a)[/itex], where [itex]\gamma > 1[/itex].

So, I can get [itex]x_{b1}=\gamma(x_{a1}-vt_{a1})[/itex], [itex]x_{b2}=\gamma(x_{a2}-vt_{a2})[/itex].

If I take [itex]t_{a1}=t_{a2}=0[/itex], then [itex]x_{b1}=\gamma x_{a1}[/itex] and [itex]x_{b2}=\gamma x_{a2}[/itex].

Now, how much distance B measures in his co-ordinate system between this two points is [itex]x_{b2} - x_{b1} = \gamma (x_{a2} - x_{a1})[/itex].

If B says [itex]L_b=x_{b2} - x_{b1}[/itex].

So, I would get [itex]L_b=\gamma L_a[/itex].

Here, [itex]L_b > L_a[/itex]. But, this should not be the case.

If I want length contraction then I have to derive it oppositely. (I found this method from http://en.wikipedia.org/wiki/Length_contraction#Derivation and http://www.fourmilab.ch/etexts/einstein/specrel/www/#SECTION14)

I have to pick some other B's co-ordinates [itex][t_{b1}, x_{b1}][/itex] and [itex][t_{b2}, x_{b2}][/itex] which is stationary in B's frame. B says [itex]L_b=x_{b2}-x_{b1}[/itex]

I have to use this equation

2. [itex]x_{a}=\gamma(x_{b}-vt_{b})[/itex], where [itex]\gamma > 1[/itex].

I would find [itex]x_{a1}=\gamma(x_{b1}-vt_{b1})[/itex] and [itex]x_{a2}=\gamma(x_{b2}-vt_{b2})[/itex].

If I take [itex]t_{b1}=t_{b2}=0[/itex] then, [itex]x_{a1}=\gamma x_{b1}[/itex] and [itex]x_{a2}=\gamma x_{b2}[/itex].

So, [itex]x_{a2} - x_{a1} = \gamma (x_{b2} - x_{b1})[/itex].

I says [itex]L_a=x_{a2}-x_{a1}[/itex].

so, [itex]L_a=\gamma L_b[/itex].

so, [tex]L_b=\frac{L_a}{\gamma}[/tex]

so, [itex]L_b < L_a[/itex]. Now, we can say this as length contraction.

But, I have started this derivation using B's co-ordinates. I as A don't know B's co-ordinates. I only know my co-ordinates because I can physically define it. I have to calculate B's co-ordinates to get Length Contraction.

I cannot use equation (1) for that, it wouldn't give me length contraction.

How can I get B's co-ordinates using my own?

If I as A has a stationary point [itex][t_a, x_a][/itex], I can calculate B's co-ordinate using my own in Galilean transformation.

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